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arXiv:2010.07819

Tensor Field Theories: Renormalization and Random Geometry

Nicolas Delporte

Published Oct 15, 2020• 1 citation
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Abstract

This thesis focuses on renormalization of quantum field theories. Its first part considers three tensor models in three dimensions, a Fermionic quartic with tensors of rank-3 and two Bosonic sextic, of ranks 3 and 5. We rely upon the large-$N$ melonic expansion of tensor models. For the first model, invariant under $U(N)^3$, we obtain the RG flow of the two melonic couplings and the vacuum phase diagram, from a reformulation with a diagonalizable matrix intermediate field. The discrete chiral symmetry breaks spontaneously and we compare with the three-dimensional Gross-Neveu model. Beyond the massless $U(N)^3$ symmetric phase, we also observe a massive phase of same symmetry and another where the symmetry breaks into $U(N^2)\times U(N/2)\times U(N/2)$. A matrix model invariant under $U(N)\times U(N^2)$, with close properties, is also studied. For the other models, with symmetry groups $U(N)^3$ and $O(N)^5$, a non-melonic coupling (the "wheel") with an optimal scaling in $N$ drives us to a generalized melonic expansion. The kinetic terms are taken of short- and long-range, and we analyze perturbatively, at large-$N$, the RG flows of the sextic couplings up to four loops. Only the rank-3 model displays non-trivial fixed points (two real Wilson-Fisher-like in the short-range case and a line of fixed points in the other). We finally obtain the real conformal dimensions of the primary bilinear operators. In the second part, we establish the first results of perturbative multi-scale renormalization for a quartic scalar field on critical Galton-Watson trees, with a long-range kinetic term. At criticality, an emergent infinite spine provides a space of effective dimension $4/3$ on which to compute averaged correlation fonctions. This approach formalizes the notion of a QFT on a random geometry. We use known probabilistic bounds on the heat-kernel on a random graph reviewed in detail.

DOI
10.48550/arxiv.2010.07819

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This thesis focuses on renormalization of quantum field theories. Its first part considers three tensor models in three dimensions, a Fermionic quartic with tensors of rank-3 and two Bosonic sextic, of ranks 3 and 5. We rely upon the large-NNN melonic expansion of tensor models. For the first model, invariant under U(N)3U(N)^3U(N)3, we obtain the RG flow of the two melonic couplings and the vacuum phase diagram, from a reformulation with a diagonalizable matrix intermediate field. The discrete chiral symmetry breaks spontaneously and we compare with the three-dimensional Gross-Neveu model. Beyond the massless U(N)3U(N)^3U(N)3 symmetric phase, we also observe a massive phase of same symmetry and another where the symmetry breaks into U(N2)×U(N/2)×U(N/2)U(N^2)\times U(N/2)\times U(N/2)U(N2)×U(N/2)×U(N/2). A matrix model invariant under U(N)×U(N2)U(N)\times U(N^2)U(N)×U(N2), with close properties, is also studied. For the other models, with symmetry groups U(N)3U(N)^3U(N)3 and O(N)5O(N)^5O(N)5, a non-melonic coupling (the "wheel") with an optimal scaling in NNN drives us to a generalized melonic expansion. The kinetic terms are taken of short- and long-range, and we analyze perturbatively, at large-NNN, the RG flows of the sextic couplings up to four loops. Only the rank-3 model displays non-trivial fixed points (two real Wilson-Fisher-like in the short-range case and a line of fixed points in the other). We finally obtain the real conformal dimensions of the primary bilinear operators. In the second part, we establish the first results of perturbative multi-scale renormalization for a quartic scalar field on critical Galton-Watson trees, with a long-range kinetic term. At criticality, an emergent infinite spine provides a space of effective dimension 4/34/34/3 on which to compute averaged correlation fonctions. This approach formalizes the notion of a QFT on a random geometry. We use known probabilistic bounds on the heat-kernel on a random graph reviewed in detail.
This thesis focuses on renormalization of tensor field theories. Its first part considers a quartic tensor model with O(N)3O(N)^3O(N)3 symmetry and long-range propagator. The existence of a non-perturbative fixed point in any ddd at large NNN is established. We found four lines of fixed points parametrized by the so-called tetrahedral coupling. One of them is infrared attractive, strongly interacting and gives rise to a new kind of CFT, called melonic CFTs which are then studied in more details. We first compute dimensions of bilinears and OPE coefficients at the fixed point which are consistent with a unitary CFT at large NNN. We then compute 1/N1/N1/N corrections. At next-to-leading order, the line of fixed points collapses to one fixed point. However, the corrections are complex and unitarity is broken at NLO. Finally, we show that this model respects the FFF-theorem. The next part of the thesis investigates sextic tensor field theories in rank 333 and 555. In rank 333, we found two IR stable real fixed points in short range and a line of IR stable real fixed points in long range. Surprisingly, the only fixed point in rank 555 is the Gaussian one. For the rank 333 model, in the short-range case, we still find two IR stable fixed points at NLO. However, in the long-range case, the corrections to the fixed points are non-perturbative and hence unreliable: we found no precursor of the large NNN fixed point. The last part of the thesis investigates the class of model exhibiting a melonic large NNN limit. We prove that models with tensors in an irreducible representation of O(N)O(N)O(N) or Sp(N)Sp(N)Sp(N) in rank 555 indeed admit a large NNN limit. This generalization relies on recursive bounds derived from a detailed combinatorial analysis of Feynman graphs involved in the perturbative expansion of our model.
We study the spectrum of the large NNN quantum field theory of bosonic rank-3 tensors, the quartic interactions of which are such that the perturbative expansion is dominated by the melonic diagrams. We use the Schwinger-Dyson equations to determine the scaling dimensions of the bilinear operators of arbitrary spin. Using the fact that the theory is renormalizable in d=4d=4d=4, we compare some of these results with the $4\ensuremath{-}\ensuremath{\epsilon}$ expansion, finding perfect agreement. This helps elucidate why the dimension of operator ${\ensuremath{\phi}}^{abc}{\ensuremath{\phi}}^{abc}$ is complex for d<4d<4d<4: the large NNN fixed point in $d=4\ensuremath{-}\ensuremath{\epsilon}$ has complex values of the couplings for some of the O(N)3O(N{)}^{3}O(N)3 invariant operators. We show that a similar phenomenon holds in the O(N)2O(N{)}^{2}O(N)2 symmetric theory of a matrix field ${\ensuremath{\phi}}^{ab}$, where the double-trace operator has a complex coupling in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions. We also study the spectra of bosonic theories of rank-$q\ensuremath{-}1$ tensors with ${\ensuremath{\phi}}^{q}$ interactions. In dimensions d>1.93d>1.93d>1.93, there is a critical value of qqq, above which we have not found any complex scaling dimensions. The critical value is a decreasing function of ddd, and it becomes 6 in $d\ensuremath{\approx}2.97$. This raises a possibility that the large NNN theory of rank-5 tensors with sextic potential has an IR fixed point which is free of perturbative instabilities for 2.97<d<32.97<d<32.97<d<3. This theory may be studied using renormalized perturbation theory in $d=3\ensuremath{-}\ensuremath{\epsilon}$.
The large-NNN quantum field theories provide a window into the regime of strongly-coupled physics. Our principal object of study in this thesis is the large-NNN family of melonic QFTs, which contain the Sachdev-Ye-Kitaev-like models, tensor models, and vector models. We begin with a review of this limit of a large number of degrees of freedom (large-NNN) as an approach to the solution of QFTs. Two toy models are used to clarify this approach: a zero-dimensional field theory and the flow of a generalized free field theory. Both models are solvable, and so we can explicitly demonstrate: using the former, the simplifications at large NNN; using the latter, the tools used to study scale-dependence of physics -- the renormalization group. We develop F~\tilde{F}F~-extremization, a simple method of solution for an arbitrary large-NNN melonic QFT in its strongly-coupled limit. The infrared conformal field theories show remarkable simplicity, in that they are entirely solved by the requirement that they have as many degrees of freedom as possible, up to a simple constraint arising from the interaction between the fields. We measure the number of degrees of freedom of the conformal infrared theory via F~\tilde{F}F~, the universal part of the free energy. We then present the example of the quartic Yukawa model in continuous dimension. This model is considered as a tensor field theory, and solved for its conformal limit; we then illustrate its multiplicity of fixed points and their stability, as well as its operator spectrum, matching the data between the large-NNN and dimensional expansions. These features reflect general characteristics of melonic conformal field theories: their existence, stability, and spectral characteristics. We conclude with future directions of exploration for the melonic theories.
Certain models with rank-3 tensor degrees of freedom have been shown by Gurau and collaborators to possess a novel large NNN limit, where g2N3{g}^{2}{N}^{3}g2N3 is held fixed. In this limit the perturbative expansion in the quartic coupling constant, ggg, is dominated by a special class of ``melon'' diagrams. We study ``uncolored'' models of this type, which contain a single copy of real rank-3 tensor. Its three indices are distinguishable; therefore, the models possess O(N)3O(N{)}^{3}O(N)3 symmetry with the tensor field transforming in the tri-fundamental representation. Such uncolored models also possess the large NNN limit dominated by the melon diagrams. The quantum mechanics of a real anticommuting tensor therefore has a similar large NNN limit to the model recently introduced by Witten as an implementation of the Sachdev-Ye-Kitaev (SYK) model which does not require disorder. Gauging the O(N)3O(N{)}^{3}O(N)3 symmetry in our quantum mechanical model removes the nonsinglet states; therefore, one can search for its well-defined gravity dual. We point out, however, that the model possesses a vast number of gauge-invariant operators involving higher powers of the tensor field, suggesting that the complete gravity dual will be intricate. We also discuss the quantum mechanics of a complex 3-index anticommuting tensor, which has $U(N{)}^{2}\ifmmode\times\else\texttimes\fi{}O(N)$ symmetry and argue that it is equivalent in the large NNN limit to a version of SYK model with complex fermions. Finally, we discuss similar models of a commuting tensor in dimension ddd. While the quartic interaction is not positive definite, we construct the large NNN Schwinger-Dyson equation for the two-point function and show that its solution is consistent with conformal invariance. We carry out a perturbative check of this result using the $4\ensuremath{-}\ensuremath{\epsilon}$ expansion.
The first part of these lecture notes is mostly devoted to a comparative discussion of the three basic large NNN limits, which apply to fields which are vectors, matrices, or tensors of rank three and higher. After a brief review of some physical applications of large NNN limits, we present a few solvable examples in zero space-time dimension. Using models with fields in the fundamental representation of O(N)O(N)O(N), O(N)2O(N)^2O(N)2, or O(N)3O(N)^3O(N)3 symmetry, we compare their combinatorial properties and highlight a competition between the snail and melon diagrams. We exhibit the different methods used for solving the vector, matrix, and tensor large NNN limits. In the latter example we review how the dominance of melonic diagrams follows when a special ``tetrahedral" interaction is introduced. The second part of the lectures is mostly about the fermionic quantum mechanical tensor models, whose large NNN limits are similar to that in the Sachdev-Ye-Kitaev (SYK) model. The minimal Majorana model with O(N)3O(N)^3O(N)3 symmetry and the tetrahedral Hamiltonian is reviewed in some detail; it is the closest tensor counterpart of the SYK model. Also reviewed are generalizations to complex fermionic tensors, including a model with SU(N)2×O(N)×U(1)SU(N)^2\times O(N)\times U(1)SU(N)2×O(N)×U(1) symmetry, which is a tensor counterpart of the complex SYK model. The bosonic large NNN tensor models, which are formally tractable in continuous spacetime dimension, are reviewed briefly at the end.
We explore in detail properties of two melonic quantum mechanical theories which can be formulated either as fermionic matrix quantum mechanics in the new large DDD limit, or as disordered models. Both models have a mass parameter mmm and the transition from the perturbative large mmm region to the strongly coupled ``black-hole'' small mmm region is associated with several interesting phenomena. One model, with U(n)2\mathrm{U}(n{)}^{2}U(n)2 symmetry and equivalent to complex Sachdev-Ye-Kitaev, has a line of first-order phase transitions terminating, for a strictly positive temperature, at a critical point having nontrivial, nonmean-field critical exponents for standard thermodynamical quantities. Quasinormal frequencies, as well as Lyapunov exponents associated with out-of-time-ordered four-point functions, are also singular at the critical point, leading to interesting new critical exponents. The other model, with reduced U(n)\mathrm{U}(n)U(n) symmetry, has a quantum critical point at strictly zero temperature and positive critical mass m∗{m}_{*}m∗​. For 0<m<m∗0<m<{m}_{*}0<m<m∗​, it flows to a new gapless IR fixed point, for which the standard scale invariance is spontaneously broken by the appearance of distinct scaling dimensions ${\mathrm{\ensuremath{\Delta}}}_{+}$ and ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{-}}$ for the Euclidean two-point function when $t\ensuremath{\rightarrow}+\ensuremath{\infty}$ and $t\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty}$ respectively. We provide several detailed and pedagogical derivations, including rigorous proofs or simplified arguments for some results that were already known in the literature.